How do I move the hair above the mesh?
It appears you are using a separate mesh for "neck hairs" instead of using vertex group of the main horse. It also appears you're using a plane.

My suggestion would be delete the plane, and use vertex groups. You make them while in edit mode, and then assign them in the particle system under "density".  Then you don't have to worry about trying to warp that plane to fit the underlying mesh during animation.

If you have to use the plane, then flip the normals, or flip the plane.